In this section, Bernstein polynomial basis is used to find a solution for the integral form of Eq. (2) as

\begin{array}{r}x(t)={x}_{0}+r{\int}_{0}^{t}[x(s)(1-x(s)-{\int}_{0}^{s}x(z)\phantom{\rule{0.2em}{0ex}}dz)]\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{x(t)=}+\alpha {\int}_{0}^{t}[x(s)(1-x(s)-{\int}_{0}^{s}x(z)\phantom{\rule{0.2em}{0ex}}dz)]\phantom{\rule{0.2em}{0ex}}dB(s),\\ \phantom{\rule{1em}{0ex}}t\ge 0.\end{array}

(10)

Let

y(t)={\int}_{0}^{t}x(s)\phantom{\rule{0.2em}{0ex}}ds,

(11)

then

\frac{dy(t)}{dt}=x(t),\phantom{\rule{1em}{0ex}}y(0)=0.

(12)

To approximate the solution, replace (11) and (12) into (10) and rewrite it as

\begin{array}{rl}{y}^{\prime}(t)=& {x}_{0}+r{\int}_{0}^{t}[{y}^{\prime}(s)-{({y}^{\prime}(s))}^{2}-{y}^{\prime}(s)y(s)]\phantom{\rule{0.2em}{0ex}}ds\\ +\alpha {\int}_{0}^{t}[{y}^{\prime}(s)-{({y}^{\prime}(s))}^{2}-{y}^{\prime}(s)y(s)]\phantom{\rule{0.2em}{0ex}}dB(s).\end{array}

(13)

Function y(t) can be approximated as follows:

y(t)=\sum _{i=0}^{n}{c}_{i}{\beta}_{i,n}={C}^{T}\mathrm{\Phi}(t),\phantom{\rule{1em}{0ex}}n\ge 1,0\le t\le 1,

(14)

where, *C* and \mathrm{\Phi}(t) are (n+1)\times 1 vectors given by

C={[{c}_{0},{c}_{1},\dots ,{c}_{n}]}^{T},\phantom{\rule{2em}{0ex}}\mathrm{\Phi}(t)={[{\beta}_{0,n},\dots ,{\beta}_{n,n}]}^{T},

so, we can write

\frac{dy(t)}{dt}={C}^{T}\frac{d\mathrm{\Phi}(t)}{dt}.

(15)

By substituting (14) and (15) into (13), we have

\begin{array}{rl}{C}^{T}{\mathrm{\Phi}}^{\prime}(t)=& {x}_{0}+r{\int}_{0}^{t}[{C}^{T}{\mathrm{\Phi}}^{\prime}(s)-{({C}^{T}{\mathrm{\Phi}}^{\prime}(s))}^{2}-({C}^{T}{\mathrm{\Phi}}^{\prime}(s))({C}^{T}\mathrm{\Phi}(s))]\phantom{\rule{0.2em}{0ex}}ds\\ +\alpha {\int}_{0}^{t}[{C}^{T}{\mathrm{\Phi}}^{\prime}(s)-{({C}^{T}{\mathrm{\Phi}}^{\prime}(s))}^{2}-({C}^{T}{\mathrm{\Phi}}^{\prime}(s))({C}^{T}\mathrm{\Phi}(s))]\phantom{\rule{0.2em}{0ex}}dB(s).\end{array}

(16)

The collocation method with {t}_{j}=\frac{2j-1}{2(n+1)}, j=1,\dots ,n+1, is used for determination of the unknown vector *C* as follows:

\begin{array}{r}{C}^{T}{\mathrm{\Phi}}^{\prime}({t}_{j})={x}_{0}+r{\int}_{0}^{{t}_{j}}[{C}^{T}{\mathrm{\Phi}}^{\prime}(s)-{({C}^{T}{\mathrm{\Phi}}^{\prime}(s))}^{2}-({C}^{T}{\mathrm{\Phi}}^{\prime}(s))({C}^{T}\mathrm{\Phi}(s))]\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{{C}^{T}{\mathrm{\Phi}}^{\prime}({t}_{j})=}+\alpha {\int}_{0}^{{t}_{j}}[{C}^{T}{\mathrm{\Phi}}^{\prime}(s)-{({C}^{T}{\mathrm{\Phi}}^{\prime}(s))}^{2}-({C}^{T}{\mathrm{\Phi}}^{\prime}(s))({C}^{T}\mathrm{\Phi}(s))]\phantom{\rule{0.2em}{0ex}}dB(s),\\ \phantom{\rule{1em}{0ex}}j=1,\dots ,n+1.\end{array}

(17)

We use Lemma 2.1 to calculate Itô integrals. By solving the nonlinear system (17), we find the unknown coefficient. Then we get the approximate solution y(t) and x(t).

The figures show the results of a numerical solution generated by the stochastic *θ*-method with \theta =0.5 and the Bernstein approximation with n=12 for two cases of *r*. The results show a rapid rise along the logistic curve and then a fast exponential decay to zero for big *r*.

They also illustrate a comparison between the numerical solutions of the deterministic and the stochastic models.